# Cyclotomic polynomial  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

In algebra, a cyclotomic polynomial is a polynomial whose roots are a set of primitive roots of unity. The n-th cyclotomic polynomial, denoted by Φn has integer cofficients.

For a positive integer n, let ζ be a primitive n-th root of unity: then

$\Phi _{n}(X)=\prod _{(i,n)=1}\left(X-\zeta ^{i}\right).\,$ The degree of $\Phi _{n}(X)$ is given by the Euler totient function $\phi (n)$ .

Since any n-th root of unity is a primitive d-th root of unity for some factor d of n, we have

$X^{n}-1=\prod _{d|n}\Phi _{d}(X).\,$ By the Möbius inversion formula we have

$\Phi _{n}(X)=\prod _{d|n}(X^{d}-1)^{\mu (n/d)},\,$ where μ is the Möbius function.

## Examples

$\Phi _{1}(X)=X-1;\,$ $\Phi _{2}(X)=X+1;\,$ $\Phi _{3}(X)=X^{2}+X+1;\,$ $\Phi _{4}(X)=X^{2}+1;\,$ $\Phi _{5}(X)=X^{4}+X^{3}+X^{2}+X+1;\,$ $\Phi _{6}(X)=X^{2}-X+1;\,$ $\Phi _{7}(X)=X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1;\,$ $\Phi _{8}(X)=X^{4}+1;\,$ $\Phi _{9}(X)=X^{6}+X^{3}+1;\,$ $\Phi _{10}(X)=X^{4}-X^{3}+X^{2}-X+1.;\,$ 